I know I saw this demo somewhere. Maybe it was at an AAPT conference a few years ago. I have always wanted to build this, but never got around to it. Until now. Here is the demo (it is easy, you should make one too)

So, how does this work? I think the simplest explanation is that the drinks do not spill because the string can only pull in the direction of the standing glasses. A slightly better explanation is that the string lets the tray rotate so that the sum of the acceleration and the gravitational field is in the direction of the open ending of the cup. I am still not happy with that. Let me start with a case where the water would spill. Suppose I just hold the cup in my hand and stop really quick. Here is a force diagram for the cup with water.

I don’t know why I broke the hand force into two components. Anyway, remember when I talked about fake forces? Basically, if I want to pretend my reference frame is not accelerating when in fact it is, I can add a fake acceleration where:

So, the diagram for the glass looks a little different if I use a fake force.

But, I really want to look at the water. Here is a force diagram for a little piece of water near the top of the surface.

I am not going to talk about the buoyancy force, but it is essentially the water below that ‘piece’ of water pushing it up. There is an interesting thing in common to both the fake force and the gravitational force from the Earth. Both pull on all pieces of the water. Compare this to the forces the glass exerts on the water. The glass can only push on the parts of the water that it is touching. So effectively, there is a new gravitational force (represented by the red vector). The water is level perpendicular to this effective gravity. If this new level makes the water go over the rim of the glass, it spills.

Now, what about the cool serving tray? I guess you can already figure out that the one-string tray adjusts itself so that it is perpendicular to the effective gravity. This happens because a string can only pull in the direction of the string. The result is that the water is accelerating, but in the direction the cup points.

This is essentially the same as the “swing the bucket of water over your head” demo, but more cooler (in my opinion). Ok, one more thing. You CAN spill the water if you are not careful. If you slip and let the tension in the string go slack, the stuff on the tray will fall. See this example.

Comments

It seems to me the most economical way of accounting for the non-spillage is in terms of inertia.

There are multiple meanings of ‘inertia’ in circulation, so first I will provide an operational definition of how I use the expression ‘inertia’.

Definition: Inertia is the amount of work that must be done to cause change of velocity of an object.

Example: an electric car with regenerative braking. When the electric car accelerates chemical potential energy in the batteries is converted to electric potential, electric motors convert the potential to kinetic energy.
When the car switches to regenerative braking the electric motors are doing negative work.

Defined in this way there is a distinction between ‘inertia’ and ‘inertial mass’. Roughly speaking, ‘inertial mass’ refers to a property of the object (the inertial mass of the car) and ‘inertia’ refers to the process that allows the electric car’s batteries to be recharged while braking.

Someone may argue: it’s the kinetic energy of the car that enables the regenerative braking.
But on closer examination: that is problematic. We have the principle of relativity of inertial motion. We can map the motion in a coordinate system that is co-moving with the initial velocity of the car (with the road speeding underneath the car). When regenerative braking is switched on the grip of the tires brings the car up to the velocity of the road.

As per the principle of relativity of inertial motion, kinetic energy doesn’t reside in a particular location, and it doesn’t reside in a particular object.

Inertia is that when two objects have a velocity relative to each other then you may harvest some kinetic energy.

Occasionally you mention what you refer to as ‘fake force’, which is an indirect way of referring to inertia. To illustrate this indirectness: it would be very awkward to suggest that regenerative braking must be attributed to a ‘fake force’.

In the case of the suspended tray we have that the equivalence of gravitational and inertial mass is at the root of the explanation. Non-equalness of inertial and gravitational mass would allow a scenario where fluid does tend to spill.